## Pre-Requisires

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**Squares and Square Roots** | **Speed Notes**

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**Square**: Number obtained when a number is multiplied by itself. It is the number raised to the power 2. 2^{2 }= 2 x 2=4(square of 2 is 4).

If a natural number m can be expressed as n^{2}, where n is also a natural number, then m is a **square number.** **(Scroll down to continue …)**

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All square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place. Square numbers can only have even number of zeros at the end. Square root is the inverse operation of square.

There are two integral square roots of a perfect square number.

Positive square root of a number is denoted by the symbol For example, 3^{2}=9 gives

**Perfect Square or Square number**: It is the square of some natural number. If m=n^{2}, then m is a perfect square number where m and n are natural numbers. Example: 1=1 x 1=12, 4=2 x 2=2^{2}.

**Properties of Square number**:

- A number ending in 2, 3, 7 or 8 is never a perfect square. Example: 152, 1028, 6593 etc.
- A number ending in 0, 1, 4, 5, 6 or 9 may not necessarily be a square number. Example: 20, 31, 24, etc.
- Square of even numbers are even. Example: 2
^{2 }= 4, 4^{2}=16 etc. - Square of odd numbers are odd. Example: 5
^{2 }= 25, 9^{2 }= 81, etc. - A number ending in an odd number of zeroes cannot be a perferct square. Example: 10, 1000, 900000, etc.
- The difference of squares of two consecutive natural number is equal to their sum. (n + 1)
^{2}– n^{2 }= n+1+n. Example: 4^{2 }– 3^{2 }=4 + 3=7. 12^{2}– 11^{2 }=12+11 =23, etc. - A triplet (m, n, p) of three natural numbers m, n and p is called Pythagorean

triplet, if m^{2 }+ n^{2 }= p^{2}: 3^{2 }+ 4^{2 }= 25 = 5^{2}

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